October 7, 2016

A bird’s eye view of topological recursion

Eynard-Orantin Theory (topological recursion) has got to be one of the biggest ideas in quantum topology in recent years (see also HERE). Today I’d like to attempt a bird’s eye explanation of what all the excitement is about from the perspective of low-dimensional topology.

Mirror symmetry is a physical idea that relates two classes of problems:

A-Model: Measurement of a “volume” of a moduli space. In particular, counting the number of points of a moduli space that is a finite set of points.

B-Model: Computation of matrix integrals.

We may think of the A-model as “combinatorics and geometry” and of the B-model as “complex analysis”. Why might relating these classes of problems be important?

Mirror symmetry might help us to compute a quantity of interest that we would not otherwise know how to compute. Sometimes enumeration may be simpler (e.g. the Argument Principle) and sometimes complex analysis may be simpler (when integrating by parts is easier than counting bijections).

An object in one model may readily admit an interpretation, whereas its mirror dual’s meaning may be a mystery. This is the case in quantum topology- quantum invariants, which live on the B-model side, are powerful, but their topological meaning is a mystery. On the other hand, the A-model invariants (hyperbolic volume, A-polynomial) have readily understood geometric/topological meaning.

Mirror symmetry (as currently understood) doesn’t in-fact directly solve either problem, but it does provide heuristics. There is no known formula to compute the mirror dual problem to a given problem- mirror duals in mathematics have tended to be noticed post-facto. Mirror symmetry is also not mathematically rigourous, so each prediction of mirror symmetry must be carefully analyzed and proven. In addition, the mathematical meaning of mirror symmetry is unclear.

Despite this, quantum topology has received a number of Fields medals for work in and around mirror symmetry, including Jones (1990), Witten (1990), Kontsevich (1998), and Mirzakhani (2014). Several of our most celebrated conjectures, such as the AJ conjecture relating a quantum invariant to a classical invariant, stem from it.

Topological recursion observes that all known B-model duals of A-model problems can be framed in a common way (a holomorphic Lagrangian immersion of an open Riemann surface in the contangent bundle with some extra structure). This was observed first in special cases, and then it was noticed that the picture generalizes. Topological recursion thus reveals a common framework to all known mathematical examples of mirror symmetry. This simplifies B-model duals to A-model problems and places them in a common framework (a-priori they are complex integrals with a lot of variables without much else in common). It also provides tools to prove mirror duality in special cases. Explicitly, all of the information of an a-priori complicated mirror dual can be recovered from an embedded open Riemann surface (plus some extra structure), whose information is again encapsulated via an explicit formula in information in lower genus surfaces. Together with Mariño’s Remodeling Conjecture, we can say that topological recursion “tidies up” the B-model side of mirror symmetry, and elucidates what it means for something to be a “B-model dual” do an “A-model problem”.

One insight which topological recursion provides is that many of the simplest cases of mirror symmetry are Laplace transforms. Perhaps this is a window to understanding mirror symmetry itself? An vague conjecture along the lines of “in some contexts, mirror symmetry and the Laplace transform are the same thing in disguise” is given by Dumitrescu, Mulase, Safnuk, and Sorkin. For quantum topologists, another insight provided by topological recursion is that it suggests ways of reframing our favourite quantum invariants, such as the Jones polynomial, as objects which have more ready topological meaning, such as tau functions of integrable systems.

So, in conclusion, topological recursion provides a common framework for B-model objects such as quantum invariants. The hope is that this will elucidate their meaning and facilitate proving their mirror duality to better-understood mathematical objects. It does this by tidying up the B-model side into something structured which begins to look tractable.

Topological recursion has already led to several breakthroughs, including the simplest known proof of Witten’s conjecture and of Mirzakhani’s recurrence, and the subject is still in its infancy. It fits well with what we know by recovering all the “right” invariants at low orders (hyperbolic volume, analytic torsion) and hitting some heuristically expected keywords (e.g. ). Topological recursion is white-hot at the moment.

Disclaimer: I’m not an expert and some things I said might be wrong- please correct mistakes, inaccuracies, and omissions in the comments!